poly-Dedekind type DC sums involving poly-Euler functions

TL;DR

This paper introduces poly-Dedekind type DC sums using poly-Euler functions, extending classical Dedekind sums and establishing their reciprocity relations in the context of modular transformations.

Contribution

It defines poly-Dedekind type DC sums with poly-Euler functions and proves their reciprocity relations, expanding the theory of Dedekind sums.

Findings

Poly-Dedekind DC sums are introduced with poly-Euler functions.

Reciprocity relations for these sums are established.

The sums generalize classical Dedekind sums and their properties.

Abstract

The classical Dedekind sums appear in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. The Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations, and are shown to satisfy some reciprocity relations. In contrast, Dedekind type DC (Daehee and Changhee) sums and their generalizations are defined in terms of Euler functions and their generalizations. The purpose of this paper is to introduce the poly-Dedekind type DC sums, which are obtained from the Dedekind type DC sums by replacing the Euler function by poly-Euler functions of arbitrary indices, and to show that those sums satisfy, among other things, a reciprocity relation.